What points in the plane can be written as the circumcenter of three lattice points?
The points that cannot be written this way have an interesting property: for any $r$, at most two lattice points are a distance $r$ away from them.
I know at least that all circumcenters are rational points, because they're the intersections of perpendicular bisectors which are rational lines.
I've also calculated a special case. The circumcenter of $(-1,0)$, $(1,0)$, and $(a,b)$ is $$\left(0,\frac{a^2+b^2-1}{2b}\right).$$
But I don't know how to tell if a given rational point can be written in that form, let alone as a circumcenter of any other triplet of lattice points.
I suppose it's possible that all rational points are the circumcenter of three lattice points, but I suspect that there are some that can't be written as the circumcenter of three lattice points, no matter how hard you try. But I don't know how I'd prove it, or how I'd catalogue them.